1 The frequency response of the basic mechanical oscillator

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1 Seismograph systems The frequency response of the basic mechanical oscillator Most seismographic systems are based on a simple mechanical oscillator really just a mass suspended by a spring with some method of detecting the mass s motions. k x(t) m u y(t) Figure. A seismometer based on the simple harmonic mechanical oscillator: mass m, spring constant k, damping constant u. Let us consider the relationship between the ground motion, y(t), and the motion of the internal mass relative to the case (and, hence, the ground), x(t). The seismometer s mass is m [kg], the restoring spring constant is k [N m ] and the damping constant is u [N s m ]. Note that the spring s restoring force is directly proportional to its extension and that the damping force is directly proportional to the velocity of motion of the mass relative to the case and damper. d 2 (y(t) + x(t)) dt 2 + u dx(t) dt + kx(t) = 0. We shall employ a linear systems procedure to determine the transient response of the seismometer to an impulsive ground motion. We obtain the Laplace transform L(f(t)) = F(s) = 0 f(t)e st dt +. s = σ + jω, j =, is the complex frequency variable of the Laplace transform.

2 of the equation above as ms 2 (X (s) + Y(s)) + usx (s) + kx (s) = 0 and rewriting, ( We obtain the transfer function s 2 + u m s + k m where ω 2 n = k/m and 2ξω n = u/m. ) X (s) = s 2 Y(s). H(s) = X (s) Y(s) = s 2 ( s2 + 2ξω n s + ω 2 n) Before determining the transient response, though, we shall explore the angular frequency response of the seismometer by assigning s = jω: H(jω) = ω 2 ( ω 2 n ω 2 + 2jξω n ω ). We may write this in polar form as H(jω) e jφ(jω) where H(jω) = ω 2 ( (ω 2 n ω 2 ) 2 + 4ξ 2 ω 2 n ω2 ) and ( ) 2ξωn ω φ(jω) = tan ωn 2. ω2 2

3 Seismometer response curve (magnitude of transfer function) H(s = jω) 0. ξ = 0. ξ = 0.5 ξ = ξ = ξ = ω/ω n Figure 2. The magnitude of the seismometer s response sensitivity (frequency normalized to the resonance ω n. 3

4 2 Seismometer response curve (phase of transfer function) ϕ(jω) ω/ω n Figure 3. The phase of the seismometer s response sensitivity (frequency normalized to the resonance ω n. So far, we have calculated the motion of the mass relative to the case (and ground) in the Fourier domain. In order to obtain a record of this motion, we need a transducer that converts the mass motion into a recordable signal. Normally we would like a signal as a voltage that could be filtered and applied to some digitizer, chart recorder or oscilloscope image. A transducer that obtains a voltage signal proportional to the displacement x(t) might be assembled as a simple photo-electric displacement device or as a simple resistive potentiometer. Another, more sensitive and less noisy design, obtains a voltage proportional to x(t) through a differential capacitor. We might describe a displacement transducer s output (voltage) signal as: v d (t) = g d x(t). This corresponds to a simple scaling of our amplitude response curves by g d where this transducer constant has units of [volt/metre]. 4

5 The most common seismometer design involves a transducer that provides a voltage that is proportional to the velocity dx(t)/dt of motion of the mass within the case. This is usually accomplished by using a magnet-coil arrangement. A magnet moving within a coil or a coil moving through a magnetic field due to a magnet obtains a velocity dependent signal in the coil. This signal can easily be amplified and recorded. We describe the velocity transducer constant equivalent to this system as g v such that the voltage obtained from the seismometer becomes dx(t) v v (t) = g v. dt You might note that a differentiation in the time domain is equivalently a multiplication by jω in the frequency domain. We easily determine the ground-motion to voltage transfer function as V v (jω) Y(jω) = K(jω) = jωg vh(jω). Seismometer response curve (velocity transducer) (magnitude of ground displacement to voltage transfer function) 0 K(s = jω) 0. ξ = 0. ξ = 0.5 ξ = ξ = ξ = ω/ω n 5

6 Note that the instrument becomes ever more sensitive as the ground displacement frequency increases. Partially, at least, for that reason, we often rather measure the output in terms of the ground velocity dy(t)/dt. We produce a ground-velocity sensitivity calibration curve: S v (jω) = K(jω) jω. 0 EM seismometer ground velocity response curve (output voltage/ground velocity) S v (s = jω) 0. ξ = 0. ξ = 0.5 ξ = ξ = ξ = ω/ω n In normal operation, the seismometer s output is filtered to remove high frequencies. Traditionally, a Butterworth filter design which provides a mirror image of the ξ = curve of the S v (jω) is used to cutoff undesired high frequencies before digitizion or recording.. Seismic noise In designing a seismograph system, we have traditionally selected frequency bands for measurement where seismic noise levels are lowest. 6

7 Power spectral density for acceleration at the Earth s surface. The upper and lower dashed curves represent the high- and low-noise model of Petersen (98). TAM: Tammanraset, Algeria In the diagram above, you might note that there is a relative low-noise band between periods of about 0.2 and 2 seconds. Instruments focussed on this region of the spectrum are called short period. Typical short-period instruments are designed for seconds bandpass. In this band, we best record teleseismic P-waves. There is another quiet band between about 20 seconds and 200 seconds. Instruments focussed on this band are termed long period instruments. IN this band, we best record surface Rayleigh and Love waves. Teleseismic S-waves tend to show periods between 2 and 8 seconds corresponding to the notable peak in seismic noise. This seismic noise is largely due to coastal ocean-generated microseisms. Now, recording with high numerical resolution of signal amplitudes, we typically try to design broad band seismographs and then subsequently filter from the record the band that we might want. The next figure shows the recording bands employed by one of the major international seismic networks, Geoscope. 7

8 The set of recording bands employed during 2000 for the station CAN, Canberra, Australia [courtesy of Geoscope]. The following diagram shows the range of ground motions (measured in terms of accelerations) that contemporary seismic installations attempt to detect. 8

9 Seismograph sensitivity to ground motions 9